ht ise 
RL TT ga 


ON A CERTAIN CLASS OF GROUPS OF 
TRANSFORMATIONS IN SPACE OF 
THREE DIMENSIONS, 


INAUGURAL DISSERTATION 
FOR {pieOw DEGREE 
DOCTOR OF PHILOSOPHY 
OUR IS 
UNIVERSITY OF J,EIPZIG. 


SUBMITTED BY 
HANS FREDERIK BLICHFELDT 


LEIPZIG, JUNE 13, 1808. 


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NEEDHAM BROTHERS 
Printers 
Stationers 
BERKELEY, CALIFORNIA. 
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CONTENTS. 


Introduction, definitions Ate 


BS 
48%. 
. The real groups similar to the eight-membered 


ye) By 


General properties and classification of the groups 


The eight-membered groups 
sroups . . - 


groups - - - 


. The seven-membered groups 


. The six-membered groups 


. The real-primitive groups in three variables 


. Some geometrical properties of the eight-membered 


PAGE. 


a Hy } 5 he A hoe 
eh, Sei i } Pea ty got ta 


“age cam 


The fundamental axioms characterizing the Euclidean 
and the Non-Euclidean Geonietries have been thoroughly 
investigated by Professor Lie in two articles, entitled 
‘‘Ueber die Grundlagen der Geometrie,” pp. 284-321 and 
pp. 355-418 of ‘‘ Leipziger Berichte” for 1890. He shows 
that a certain class of Continuous Groups of Point-trans- 
formations is intimately connected with this question, and 
determines such groups tor space of three dimensions ; the 
groups being defined by the following properties: two 
points have one, and only one, invariant ; s > 2 points have 
no invariants independent of such two-point invariants. 


Following the plan of these articles of Prof. Lie, the 
writer proposes to determine and investigate some of the 
finite groups tn space of three dimensions, for which not less than 
three points possess invariants, and for which s points, s > 3, have 
no invariants independent of such three-point invariants. We 
shall consider only those groups whose transformations are 
analytical functions of the co-ordinates x, y, z, as well as of 
the occurring parameters. * 


T) ‘‘Berichte ueber die Verhandlungen der Koeniglich Saechsische 
Gesellschaft der Wissenschaften zu Leipzig. Mathematisch- 
Physische Classe.’’ Vol. 42. 


2) To this class of groups belongs the group of Euclidean 
Motions and Similar Transformations : 


DMM tain We Sh yees we. OF 
mane sae Meeiigiy OX ner OR OY J Ox 
Cae o7 OF 

ee Bey eee 

Two points have no invariants for this group, but three points 


have two, namely, two apgles of the plane triangle which has the 
three points as vertices. 


Sah 


The notation is that of Prof. Lie. A group formed by 
p independent infinitesimal transformations 


Xf = 8 (a, 9,2) Ltn oy, at +6 (2 Kaye 


we shall call a o membered group. ‘The term ‘‘ Bahncurve’ 
we shall translate by ‘‘ pathcurve.” By the ‘‘ combining” of 
two infinitesimal transformations X;f and X,/ we shall 
mean the forming of the ‘‘ bracket-expression”’ (Klammeraus- 
druck’’): 


(XS; Yaa) =e OU ae X Gye 


A. GENERAL PROPERTIES AND . CLASSIFICA~ 
TION OF THE GROUPS. 


The number of independent three-point invariants for any 
one of the groups considered can not be greater than 
three, as is easily seen from the following consideration. 


Let the ¢ infinitesimal transformations 
; oF 
Xf ee Gi (ney, =a) ve ae th rn z) a see % (x, 4, 2) 5 Se 
as) sO) ee 


form a group with the required properties. We have then 


p 
(Ae hn) 2 Cis Aa Jie alee oe 1 oe 
Cx. being a constant. 


Theinvariants of the three points 2,,/y,, 2,4, Vj, 2) 0a 
g, are the solutions common to the partial differential equa- 
tions 


We fa XO PE XO f+ XO f= ol i =, 2, ee 


where 


Age i (ea, Vny By ee. 1); (a Dny a) 


of 
OVn 


Beer (ay Vay 20) Ue Hae Ory, 
Zn 
Now, were there more than three independent solutions, 
the number of independent equations would be less than 
six. In this case then, there would certainly be less than 
six independent equations of the system 


ee XO fA X, OF So ees a: 
written in the six variables 7,, y,,.2,; %,, y,, z,. Now, these 
equations form a complete system of partial differential egua- 
tions, 1. e., for every value of the subscripts 7 and &, 


igs oleae 
aus fi 


i 


(Wis WS) 


They would accordingly have at least one common 
solution, which would be an invariant for the two points 
1, Vy) 25 Uy, Vy 2%) 5 —Coutrary to the hypothesis that not less 
than three points should possess invariants. The assump- 
tion that any one of the groups considered has more than 
three invariants for three points is therefore false. Hence: 


The groups under consideration fall into three classes, 
according as the number of the three-point invariants ts 
one, two or three. 


There being no invariant relations for less than three 
points, the groups must be two foid transitive. There 
will therefore be three conditions imposed upon the indepen- 
dent parameters of such a group to fix one erbitrary point in 
space, and stx conditions to fix two arbitrary points. 


SEs aad 
Consider now the first class of groups, where three 
points are connected by one invariant relation, say 
Zi (7, Jiy 2 ) Vos Vay zy ; Vs, V3) 2.) —— constant 5 
Hs, Vis 25 Vas Vos %n3 Te, Vos 2% DEIN the co-ordmatessocmmrae 


three points. 


If by the general transformation of such a group, the co- 


ordinates +7,,.,,°2, 3°", etc: take the values’ 6.’ (oy, Gee 


etc., we have the relation 


vi (",, Vy 43 %ay etc. ) = fats vs Ce — etea), 
or, as we shall for brevity write it, 


L553 223)3) ee) ee BMD Fi ik | 


Let us now fix two arbitrary points, as %,, y,, 2,3 2, Vo).% 3 
[foots ie ee am Be a yl ee 
so that @, == 7, == 9,32, = £6 eee 


The number of independent parameters of the group is 
thereby reduced by stx, as we have seen above. 


The relation (1 ) now becomes 


LEGER 8) ToT ee 3) ES a ee eine ee ea 


showing that any, other point as +,, ¥,, 2, in space is re- 
stricted to move on a surface. Two conditions more im- 
posed upon the parameters of the group will therefore fix 
this point. The equation ( 2 ) then becomes the identity 


ER Mak aw a oe ae 


The motion of any other point as 2,, y,, 2, is now re- 
stricted by the relations 


Jil Dead) oa oN aR Rp 2, 4’), alia, 3) 4) = hoo. a 4’), 
Pay3,4) = 2 (203, 49s ee 


Oe 

If it is impossible to eliminate x,, y,, z, from these equa- 
tions, then this point x,, y,, 2, cannot move in a continuous 
manner, and we shall regard it as being fully fixed. In 
this case therefore, the group is reduced to the identical 
PeaMaioeiialOn et 2), =), 2, ==2,,' by Imposing 
upon its parameters 6 + 2 = 8conditions. Zhe group ts 
accordingly eight-membered. | 

On the other hand, if it is possible to eliminate x,, y,, z,, 
from the three equations (3), but not from any two of them, 
as. 

Wee yom, (1,2 4") 3-2 (1, 3, 4) = 7 A 33 4), 
then this point x,, y, 2, can still move on a curve. One 
condition more will therefore suffice to fix it, and it is evi- 


dent that any fifth point x,, y,, z, being bound by the 
independent relations 


Pe 25) 4115 24/5), /:(1, 3) 5)=2Z (13,5), 
Were, 5) = / (1, 4,.5') 
would be fixed also. This group is therefore 6+24-1=09- 
membered. 


5) 


In the case where only one of the equations (3), as 
eo) =) ie, 4), iS. independent, itpis, clear, that 
dfter fixing any number of points as %,, y,, 2; etc.; ey. Vers 
Z,,, the relations restricting the motion of any other point 
Ky Ves By 


Mee a Ca, UNK), A) 0 Sarl ee eed 
would be equivalent to only one, 


Uae) =n oeienl 


so that this point +, x, 2 would still have two degrees of 
freedom. But, as we are only considering fzzte groups, 7.e. 
groups with a finite number of arbitrary parameters, we 
easily see that this case must be excluded. 


—-[o-— 

A group belonging to the first class ts thus produced by 
either nine or eight infinitestmal transformations. 

Consider now the second class of groups. Here a group 
has two invariants for three points, say 
vi (V5 ays Xo; Zo} V3, V35 24) VAR Oar 21,4 Coy y 2) Bos X3, Voy 2.) 
which we shall for brevity denote by 

Z (1,2, 3) are 3) 

Four, points%,, 9), 2,3 ->.'.3 4, 2, have theming 
7 (1, 2,3), 7(1, 2,4), 21, 3,4), 7 (2, 3,4), ) 
J (1, 2, 3), J (t 2,4), J, 3, 4), J (2, 3,4), 


of which at least the following 


£'(1,52,-3)) Ps 2y-4) ef (L230 Ly ee) oe rr 


would be independent. 


(4) 


Now, could the remaining invariants of the system (4) 
be obtained from these, all the invariants of s points would 
be obtainable from. 7 (1, 2, 7), \/ (is 2.2). 4—=3, 4, rr 
That is, we should have but 2s — 4 independent invariants 
of s points; in other words, there would be only 2s—4 
independent solutions of the system of partial differential 
equations 


We f= ROP pK Of rete ee 


1 ST SR an eS 
where 
a eee (tate ee erry aay 9 a 
2 ) 1) n 1 ny ns Bn 
dX, o 
= ‘¢ ea vny Ynys Zn): 26 
21 
X, f, (== 1; 2) + 14) Pp being ‘the: anfinitesimal): transis 


tion of the group considered. ‘Taking s sufficiently large, 
the equations of this system would be independent of each 


other, and their number po would accordingly be 3s — 
(2s—4) s+ 4. The group having a defincte number of 
parameters, we conclude that the system of invariants (5) 
is not sufficient to determine all the invariants of the four 
points. We must add to the system at least one invariant 


more, say /(1, 3,4). Then the following invariants of 
Ss points : 


nears) ye /-( Ly 2 753.0 2 (ary Ofna me MTD ve it Ine auege be 
i as A 


in all 35—- 7, are easily seen to be independent of each 
other, and therefore 


Pi Sn8)S (3 San 7)) et 7 


Moreover, since three points have two invariants, 


Pee Sie 357 276 Ol Pee. 


The groups of the second class have thus seven tnfinttest- 


mal transformations that are independent. 


We come now to the groups of the third class. Let us 
denote the three independent invariants in the three points 
PO VE Ao kah i ceee ey a eae by 2) (tore. 3), Pi Aie ee ewe 
my 2 eis 

s points have plainly the following independent invariants: 

TAD? NOT iit) thoy Th) Seto Us geil) sn nadie oh A gos, 6, 95; 


the number of which is 3s—6. Hence p = 3s — (3s—6), 
pe tO: 


Moreover, three points have three independent invariants, 
and therefore » = 3.33, or p = 6. 


The groups of the third class are accordingly six-membered, _ 


We saw that the groups of the first class are either nine- 
or eight-membered. In the first case, let 


Va 
BE yo Sj (ae; J) 2) ee = eth (A Vs z) <n 
+ € (x, ay, ig) ue i= 1,2) 


be the infinitesimal transformations of such a group. 


In order ‘that three pointsic,, 4,2; *7 92). 
may have an invariant, but eight of the nine differential 
equations, 


Wf = XO F + APS + 4 = 0) ay ee 


can be independent. Let these eight be 
GE Pi ace Meret 8 SS SY 


Now, the equation 

VS EXPOS XOL + XOL + XPF= 0 
cannot be a mere consequence of the eight 
Vif se XO Ff EXO GS AOS + XO = 0) f= 1, ee 
unless we have a relation such as 


8 


; - on saa 7 (4) <= 
a ae (25 JV» 713 Va, Voy 225 Vy V5 2) Aj '/ oo O, 


which is plainly impossible so long as X,%f X,f, ete. 
Xf are independent infinitesimal transformations, 


m, 


Hence, the differential equations 


vi= x fee Dae ieee XO f+ DOE Gp a a ONES: Gh 


are all independent. Four points have therefore only 
4.3—9—3 independent invariants, so that 


f (2,3,4) =F {7 (1, 2, Bh lees Aaa (E: 3) 4) § oo ge ay (6) 


If we now fix the point 2,, y,, 2,, the resulting six- 
membered group will be of the kind considered by Prof. 
Lie in the above-mentioned memoir ‘‘Ueber die Grundlagen 
der Geometrie.” For, it appears from (6), that any inva- 
riant 7 (a, 6,c) of the group can be immediately deduced 
from invariants of the form J (1, a, 4), which become 
merely two-point invariants when the point x,, y,, 2, is fixed. 


Suppose the nine-membered group in the first place had 
an invariant system of surfaces, say + = constant.” ‘Then 
the group would be of the form 


KS O)L + man VZ to ana, 
eT 2 hers U. 


We know from the theory of continuous groups in one 
variable that the shortened group (‘‘verkiirzte Gruppe’) 


renee Bes Wi odie 1 15% 


1) We say that a group “as an invariant system of surfaces (or curves), 
or that such a system Jdelongs to the group, when the members of that 
system are interchanged by the transformations of the group. 


is at most three-membered. After fixing the point x,, y,, 2, 
it is therefore at most two-membered. But, Prof. Lie shows 
in the above-mentioned memoir ‘‘Ueber die Grundlagen der 
Geometrie” that, in the case of the six-membered groups 
there considered, if they have an invariant system of sur- 
faces + = constant, the shortened groups 


must be three-membered. We therefore conclude, that 
the nine-membered groups can have no tinvartant system of 
surfaces x = constant, 


There might be an invariant system of curves + —con- 
stant, v—constant, hut no system of surfaces, connected 
with one of the nine-membered groups considered. In such 
a case the group must be of the form 


ears, oF ee On i 2 
A= oF (x, 9) E+ 1; (myo f (x, 9,2) SE, 
i= NR ees ahs Gis 


the shortened group, 


xX 


KP gee Mh (x, 9) 2, 
3 oy 


being priniitive. 


Now, all such groups have been determined by Prof. Lie.! 
But, before we go on to consider these groups, it will be 
necessary to extend a remark made by Prof. Lie in con- 
nection with his articles ‘Ueber die Grundlagen der 
Geometrie.”’ He shows that, in case of the six-membered 


1) See chapter 8 of his work ‘‘ Theorie der Transformations- 
Qruppens i) Vole 


groups, no two infinitesimal transformations can have the 
same pathcurves.' In our case that becomes: in all the 
groups we are considering, for which not less than three points 
have invariants, and for which s points, s > 3, have no in- 
variants independent of such three-point invariants ; —no three 
mnjinttesimal transformations can have the same pathcurves.” 


Suppose there were three such tranformations, X,/, X,/, 
X,/, so that X, f=, (1y2,) HS ASHP; (4,2) GS 


The partial differential equation 


r ¥ of of (Laat of = 
—— Cc. 4 Sass ) — ——- J — 
Oe es E(2I,2) i (x,y, aes r Sila, 9,2) 0 


has two independent solutions, wz, v, say. Let us change 
the independent variables in the group to x,w,v. Then 
Xf, X,fand X, f will take the forms 


Me ST ae ale 
cd er it, v) ox’ L, 0x Ys Ox 


Meniuvatialis  Oimime three: pOomts 144. °24,, Vy jo vy, dace 
X,, U,, V, are determined as the solutions of a number of 
differential equations, among which we should find the 


following : 


1) See page 371 of ‘‘ Leipziger Berichte ’’ for 18go. 

2) It will be noticed that, in the general case where a group inz 
variables has no invariauts for less than m points, the invariants of 
s >m points being all dependent on the m-poliit luvariants, 20 m in- 
finitesimal transformations can have the same pathcurves. The proof would 


_be similar to that given above. 


— {1 6h— 


be 


ip, a + 90h Brae we=0 > adie ee eee Oar 


sage " ot Mor (3) af —— | 
p,. Bert a ERA, ean J 


The determinant 2 + #,'?. 7%, .¢,° could not vanish 
unless the relation 


a + 8.09 + 7G? = 0 


were satisfied, the co-efficients a, f, y, being functions of 
Gh es as ] es: 1) 
Hn Uy) Uy} Agy Uy Vqy Oly. ~ Phe quantities 200 ers aaa 
are, however, functions of +,,z,, v,, only, and accceainee 
independent of x,, 7,3; 23, %¢s,V, Hence a, 6, y should, 


be constants; but this would lead to the relation 
aX, f A: A Agha 


contrary to the implicit assumption that X,/, X,/, X,/f are 
independent infinitesimal transformations. 


Since therefore the determinant 2 + o,% 7,° ,® is not - 
zero, the equations (7) are equivalent to the system 


) | ) 
somo. =o, = =o. 
The itivariants in the three points %,, 4, 0, ; 4.7 
2X4, U,, V, would accordingly be free from the variables 1,, x,, %,. 
Now, the invariants of s points, s > 3, are all, according 
to hypothesis, made up of the three-point invariants, and 
would thus be free from x,, %,,...., 2. Among the differ- 
ential equations defining these invariants we should then 
necessarily find the following : 3 


cs ap aa eae os Dei ecen coe eat tea 
s independent equations in all. But, the group considered 
being finite, containing p infinitesimal transformations, say, 
it could not give rise to more than p independent differential 
equations in any nuiimber of points. By choosing s suffi- 
ciently great, we see the impossibility of the system (8), 
and our assumption that there could be more than two in- 
finitesimal transformations with the same pathcurves is 
contradicted. 

We now go back to the nine-membered groups of the 
form 


fas A (x, 9) E+ Nea 


MRERCLE  oipeke Ane 


No three infinitesimal transformations may have the same 
pathcurves, and all the nine differential equations 

ee hee fs Xf Gy fe, 
must not be independent of each other. 

Referring to the work by Prof. Lie” on groups in three 
variables, we find without difficulty that no nine-membered 
groups of the required form comply with the conditions 
given above. 

There are no nine-membered primitive groups in three 
variables.”) Thus we see, that no nine-membered groups 
satisfy our requirements, and the three classes of groups are 
therefore respectively 

exght-membered,; one invariant for three points, 
seven-membered, two invariants for three potnts; 
six-membered; three tnvariants for three points. 


1) ‘‘Theorie der Transformationsgruppen,’’ Vol. III, Chap. 8. 

2) See Vol. III, Chap. 7 of ‘‘Theorie der Transformations- 
gruppen.’’ Prof. Lie had already given the primitive groups in 
“Archiv for Mathematik,’’ Vol. 10, 1885. 


—T3— 
B. THE EIGHT-MEMBERED GROUPS. 


There are no primitive, eight-membered groups in space 
of three dimensions.” 


If a group belonging to this class is of the form 
KSEE (Lt mle nL Gay, De 
Uma Yo 2, SRA Mah 


the shortened group 


Paes aees 


must be three-membered, otherwise we should have one or 
more invariants in the three points 7,,,,2,3 Xo. V2.3 Xs) Var 2 
involving x,, x,, x, only; namely the solutions of the differ- 
ential equations 


Wife E(x) + &, ( 4) + Ela) ee =o, 2—=T1,.. 


The invariants of s points would therefore contain only the 
co-ordinates. 4,,:2%,5..: +). 4% But) then the  dittemesacs 
equations determining these invariants would be composed 
at least of the following : 


2 ae a ki Of Ot Ueno 
Bye By are he, 
OF 
Aote e 


in all 2s, This is impossible, since the total number of 


these equations can not be more than 8. 


1) See Vol. III, Chap. 7 of ‘‘Theorie der Transformations- 
gruppen.’’ 


. All groups of the form 
wo ; of ne 
ASE &i (1,9) S-+ mx) talane, is to ee aie 
and the shortened group 
XH ale + n(x, n¥ 


being primitive, are, as before menuoHea: already deter- 
mined, as are also the groups having a single imvariant sys- 
tem of surfaces, += constant, but no invariant system of 
curves of the form + —constant, y= constant. Only the 
following two types are eight-membered and have no three 
infinitesimal transformations with the same pathcurves : 


Bik eh ition eee 22? P27, xp ya, 
Ce ety eee Aah AD 3% feel yt ae 


B P) G7, 9; ADIGA YP, ed =, 
ep eg x IP 7 A/T. 


(Here we have for shortness written /, g, 7, in place of 


f of of 
O07, Oz, 


respectively, which we shall largely do in the following 
analysis. ) 

It remains now for us to determine the groups having an 
invariant system of surfaces x=constant, and having be- 
sides an invariant system of curves of the form +—constant, 
y=constant. Such groups will be of the form. 


Xf= &(a)p + mx,v)9 ae A Vie, eT 2, = aS 


——70)—-— 
We have seen that the shortened group 


must be at least three-membered (p. 18). Moreover, the 
shortened group 


X f= El(a)p + n(xy)¢ 


must be at least six-membered; otherwise we would have 
three infinitesimal transformations of the form 


(a y,2)7, 


which would have the same pathcurves. 
Now, all groups in two variables x, y, have been deter- 
mined by Professor Lie.”» For the required shortened groups 


iy (a= Ei(x)p —|- nlx, V)9, eS Lye eis oem 6, eS 
we find the following types: 
The group Xf eight-membered. 


q 9, x9, xg, x'g, p, xp 2V9, xP > 4xy¢. 
Gi XQ; BQ)" KO, 99; Pr Zep 3g, ee 


The group Xf — seven-membered. 


G3 XG, 29, XG, Pp; 2XP 7 399, Gp 1 310g. 
G49) BINT, PP VG ee en ae 


The group X:f six-membered. 


G5 KG = iG SP; AP AYO, (ape we yg 
G XQ, IQ Pp, 2XP + 9, XP + XV. 
INGE I a Pata, aot 


1) ‘‘Mathematische Annalen,” Vol. 16. 


—- 2 ik —_— 
The eight-membered group Xf. 


The shortened group X, / has here either of the forms: 


TAG ed, OG Pip 2y9, 2p -- 4x9; 
Bee edt, V7 Py 2rP + avg, 1p + 3cy¢. 


We have now to determine functions QP, eter on 
x, y, 2, so that the two sets of infinitesimal transformations: 


Xi f= 9+ ar, f= x9 Pr, Kf xg + yr, 

AG =aAg + 07, X, f= p  &, Xef = 2xp + avg + ir, 
XSEH XP + 40g Ur, Xe f= xg + Ir; 

MSH qt ar, Kf x9 fr, XY f= xg + yr, 

X f= x09 + Or, Xf=p + tr, Xi f= 2xp + 3ya + 1, 
Xi f= xp + 3xvg Mr, Xf yg Ar; 


form groups; 7. é. the relations 
8 
XS, aay) = 2s Paisthel pies BeSRR Shee of 8, 
1 


where ¢,,, is a constant, must be satisfied. 
- Both groups have infinitesimal transformations of the 
form 
gtar, x¢q+ Br, eg+t yr, x¢+ or, p+ e&, 
2xp + kyg +, p+ kaya + ur, k = 4, or 3. 

For the variable 2 we may substitute any tunction of 
x, y, 2, provided this is not a function of x and y merely. 
It is then plain that by the substitution of 2’ for z, where 
2’ = P (4, y, 2) is a solution of the differential equation 


we change p+ & into f, without at the same time altering 


the form of the other infinitesimal transformations. Having 
done this we write z for 2’. 
Then from i 


gt eee 


we see that 
Oa ae 
Ox ana? 


since we can have no- infinitesimal transformations of the 
form or. , 

Thus g+ av=q+/A(%2)r. We can now find a 
substitution 2’ = @, (y, 2) (where @, is not a function of y 
merely), which will change g + /, 7 into g and leave f un- 
altered. We then drop the prime, and we have thus deter- 
mined the infinitesimal transformations 


Pp: q. 


Combining these in turn with +g + fr we get 


a 33 
(p, XG, = OC) 7 Oras erie 
6 
eatin hy 
Hence 
OG Bet 
Ore HeeOy. Sakae 


so that +g + fr 1s of the form 


Bg ae he Oe 


By a change in z alone we can cause f(z) to become a 
constant, a, say, which is either o or 1. (In the following 


analysis we shall denote constants by the letters a, 4, c, etc.) 
Again, from the bracket expressions 


Py 24D IG OT) = fp ae, 


02 
Gy 24p A RG 4 7) = kg sy me 


(49 + ar, 2xp + kyg + 1) = (k— 2)ag + & : : | as yr, 


we find that 1: = /,(z), when @ = 0; and 1 = (£—2) z+ ¢, 
when a = 1. 

Let a=o. In this case we have obtained the infinitesi- 
mal transformations 


Pr % %9, 2xp + kyg + flzyr. 


Combining g with 2° + kayg + ur we get 


: (Qa Pp i RAVG + ur) == kag + care 
Hence, ; 
GU Te 
ceo O. 


Then from 


0% 
7 


(19, 2) + kayg + ur) = (k—-1) 29 1 ry ae 


we obtain the infinitesimal transformation +g. Now we 
have three of these with the same pathcurves, 


Peet Ligs 


which is not permissible. [he case a = o is thus excluded. 


Sieg 2 


Next, let a— 1. In this case we have reduced to 
simplest form the infinitesimal transformations 


BG 9 11, 2xb + kyg + R+ 2) e+ 0/7. 


Since £ is not equal to 2, we can change 2 so that ¢c disappears. 
Let us combine /, g, and 2x%p + kyg + (k — 2)zr in turn 
with 274 + kayg + ur. 


‘The results are 


2xp + kyg + oy, sae +- ve 2x°p +- 2kayg 


tp Dats b/s 6 
t Gas 4 ys +k 2G oe 


Each of these infinitesimal transformations must be of 
the form 


l(aqg-+ 7) + m (2xp + kyg + (k — 2) 2r) 
+n (xp +t keyg + 47). 
Hence 


nu — (k 


ote Rye 


Then from the bracket expressions 


ag 7, XP + kayg - ((k — 2) x2 + ky) 7) 
= (k — 1) (#9 + 247), 
(x'q + 2nr, x'p + kayg + ((k — 2) xz-+ ky) 1) 
= (k— 2) 9 327), 
(x"q + 3¥'7, XP kayg + (Rk — 2) ay + hy) 7) 
= (k — 3) (9g + 42°7), 


we get the remaining infinitesimal transformations of the 
group where & = 4, which is thus given by: 


py g, xgtr, 2g + ear, 89+ 327, x'¢g + 427, | 
ap -- 2yq + ar, P+ 4nyg + (2e2- 4y)r. | 


In the case of the group where &£ = 3 we have yet to. 
determine A in yg + Ar. 
From 


or 
(PEUR An) eT; 


oA 
@, 99 + ANv=ot+ ie 
we find that A = f(z). | 
Then from 


3 
(ag +7, v9 + fr) = x9 + ey 


(xp + ay9 + er, 9 + fr) = CE — for. 


we finally see that /,(z) = z. 
The group has therefore the form: 


Pd, xa +47, ¢ + 2xr, x2¢ + 327, vq + 27, 
2xp + 3y9 + ar, xP + 3xyq (eZ + 39) 1. 


The seven-membered group Nh 


The types of the shortened PrOUp TAI fe U el, Bo hairs) 
are here the following: 
Q XG XG, ©, P, 2XP + 3IG XP + 3479; 
9, ~9, 9, V9, p, 2XP + 2g, KP + 2x99. 


Besides adding to each of these infinitesimal transforma- 


eae ae 


tious of the shortened group a term @; (4%, y, 2) 7, we must 
add an infinitesimal transformation Av in each case to com- 
plete the groups. We can evidently put A = 1, which is 
equivalent to changing z. The groups are thus of the form 


KIN, F Apne? Sepa + 


Now, from 
(7, Xf + Pr) ae 


we see that 9; (4, y, 2) =az + VY (x, »). 
‘Then from 


(Xft faz+ Wir, Xft faz + Win 
== (Ko, Xf) + Mey, 


we easily see that the infinitesimal transformations of the 
‘‘derived groups’”’ do not contain z in the coefficients of 7. 

The derived groups of the above-written shortened groups 
Rife Tao hag op a heve an common the infinitesimal 
transformations 


BP 7; %9, #9, 2xp + kyq, 6 + kxyg, k = 2, or 3. 


The complete groups have therefore in common infinitesi- 
mal transformations of the form 


Porar, g+ fr, xq+ yr, e¢g+ 6r, 2xp + kyg + &, 
Xp A kxyg@ iy rs hk 2, OF == 3 


the functions a, /, etc., not containing 2. 
We can at once put a=o. Combining then f# with 
q + Br we get ; 


07 
(p. q+ 67) = Er, 


which shows that 


asa ihe 
pcre 
so that 
p Se + f(y). 


By a change in 2 of. the form z= 2’-+ F(y) we can 
cause f(y) to disappear without at the same time altering 
the form of the infinitesimal transformations other than 
g + fr. , 

We now have the‘infinitesimal transformations putting 
2 for 2’) 


By G+ axr. 
The groups have now the forms 


Yr, p, g-axr, xq+yr, q+ 67, 2xp+ 3r¢g + &, 
xD oa 3XVG +e 17’, KG + 8 


CORO a ieee Veg Or, .2np apg fF, 
wD 209g + 1, YE UT; 


where y, 0, € and z do not contain 2. 

Then by forming bracket expressions, etc., we find with- 
out much difficulty that the following group only has no 
three infinitesimal transformations with the same path- 
curves : 


| 


| 


E LEP GOATS AE OIG A ei sep a VG, 


xp + 2kyq yr 


hee 
The six-membered group X,f. 


The different types of the group Xf, 7 = 1, 2, ...-. Os 
are as follows: 


FAQ 49, Py 22 pc IG Pr Oey: 
9X9, IQ P, XPT VG *P 1 IG; 
I VG: I Uo: Pr AP EL: 


Each of the complete groups we are seeking will contain 
two infinitesimal transformations of the form a(1%,y,z)r, 
f(x,y,2)r. | 

We can evidently put 6 = 1 at once. If (7, ar) does 
not vanish, then, according to Professor Lie, we can so 
choose the variable z that these two infinitesimal transfor- 
mations take the forms 7, 27. 

Combining them with any other infinitesimal transfor- 
mation 


&(x)p ae MXV)9 a o(4,9,2)7 


of the group we get 


e 


(1, \G(2)) + eo SC 2 a er, 


Che E(x)p ae MXI)G ne C(4,y,2)7) ce (eS a o)7, 


which gives us the equations 


oO 
Taira 


d€ 
=a + bz, 2— —C =a + vz. 
OZ 
Hence, € = 4 -+ #2, which result, for our purpose, is | 
_ equivalent to € = 0, since we already have the two infini- 
tesimal transformations 7, 27. 


We therefore simply add 7 and zr to each of the shortened 
groups X; PI oat TD haa sacs , 6; but then it is easily seen that 
in each case there would be three infinitesimal transforma- 
tions with the same pathcurves. 

We need therefore only consider the cases where 


Tar keaa Oo: 
so.that a= f, Gr, 9). 

Let Xi f+ —, (*, v, 2)r be any infinitesimal transforma- 
tion of either of the groups we are considering, the corre- 
sponding infinitesimal transformation of the shortened 
group being X,f Combining this with + and fA (4, y)r 
successively we get 


OD | OD, ener 
og Y) (ise eR X;fi)r. 


Hence, 


i , , dD ert 
ee ea — Xf = A’ a mee SE 
giving Eee 
Rh = a+ i+ of 


Two of the shortened groups Bye a 1 a eas , 6, contain 
g, xg. Accordingly we should have in those cases 


Yea, + bf + Geka wet a bhatt 


from which equations it would follow at once that (. would 
be a function of x alone. 

The same two shortened groups contain ~, 2%p + yg, 
Moa LOT 2. 

Therefore, 


a a 4 fp eft sgh — at BAL Cif 


By means of these equations we find without difficulty 


that : ‘ 
ee 
Now change z into 


whereby 7 is changed into (m«-—+ x)r. By this change we 
would get the infinitesimal transformations (mx -+ z)r, 
(hx + l)r, instead of 


hx el 
* i 
mx + 2 m 


These we can replace by the two, 
Pea 
since they must be independent. 


The shortened group not yet considered is g, yg, y’¢, p, 
xp, x*p. From the equations 


OF, 2 Off 2 E : 
By TST Cal 9 D5 ore ae aa 


we get at once 


He ans l 
f= Ke, rps ere 


Again, from 


Fama + A+ eff, 52 = A+ BLA CR 


we get fe 
eee we 


Sine 7 pea fk 


It is plainly sufficient to consider only one of these cases, 
since we get the other by interchanging x and y. 


We shall therefore, for the two infinitesimal transforma- 
trons, av, fr, use the two 7, xr. 


Two of the groups we wish to determine have in common 
infinitesimal transformations of the form 


Lely PA V7, Gi} Ov, oag 4A ery, BAD TT RVG aS OY, 
2p + kaxyg + ur; k= 2, 0r = 1; 


while the remaining group is of the form 


r RIDES ME, PO, WG a ap yD 
TM AMES ame 


By combining any one of these infinitesimal transforma- 
tions with 7 and «7 we find without difficulty that the 
functions y, 0, €,1, and A are of the form az + @ (x, y); 
and that x= (6+ 4)2+ 4% (x, vy). Then by building 
bracket-expressions, and by choosing suitable functions 
z+ f (x, y) to replace 2, we finally obtain one group of the 
form required, namely: 


| vr, xr, p, 9, x9, v9 + (ay + bz)r, a (6 — 1) = 0, 


2xp + yq+2r, xb + xyqg + x2zr; bis not equal too. 
| 


The eight-membered groups are now completely deter- 
mined. We have the following types: 


Pp: gq, +9 = Y, Xp Sh oe ae Es yp — eases xp = VQ; 
vp + xyqg+ (y — xa)r, ayp ty’¢ + 2 ly — ¥a)r. 


Te | Pp» J, +79; xp paras 6 Di, <p + vq + YT; 
| KP + yg +77, xP IE IT 


Pg x97, Hg -- 2x7, x9 + 3x'r, x19 + 42°F, 


r. | 
xp + 299 + 2r, xp + 4ayg -P (2x2 + 4y)r. 
. Di G49 7, 29. 247 kG 4 327, Soe 
xp + yg, ep + 3x9q + (x2 + 3y)r. 
: Vy Pi Tr AQ; BL TIS IY AT aD, ae 
_ xp 2xVq Yr. 7 
7 0 0 Bs Gs 0s CAPT Id ar 7d 


ap + xyq-+ «zr; 6 is not equal to 0, @(6— 1) ==0; 


The question then arises as to whether these groups 
possess the following properties: 
Zr. There is one and only one invariant of any three points. 


2. The invariants of s points, s > 3, are not independent 
of such three-point invariants. 
The first requirement is fulfilled by any one of the groups 


Myf GAO Yep a Mikes WC oi ieee 


Dian Cae a seat 
if the eight linear partial differential equations 


Xf Ay f DG Sag jae 0, 2= E325. ieee yo 


where 


XP = Er, 16) + nena) + Gn Set a 
are all independent. We readily find that they are inde- 
pendent in each case. 

We shall now consider the second requirement. Let us 
for brevity denote the invariant of the three points ,, 1,, 2,; 
Ay Sok Ent Hes Voy Ze BY L (a, b,'c}. 

If the determinant 


d/(1,2,4) d/(1,3,4) 0/(2,3,4) 

dle OX, : OX, OX, 
o/(1,2,4) d/(1,3,4) d/(2,3,4) 

OV OV OVs 
0/(1,254) 0Z(13,4) 8/(2,3)4) 

024 ; O24 . 02, i 


were different from zero, then it would be impossible to 
eliminate the variables +,, 1%, 2, from the invariants 
Tt. 2A) ee onan), /. (1,3; 4); eenerally speaking, it 
would be impossible to eliminate »,, y,, 2, from / (1, 2, a), 
7 (1, 3,2), 7 (2, 3, @); and therefore the invariants of the 
system 


Pp sie) Rsk WAG aire Caaria een (Owe TXD F 
Urea hy MUL eaed PLCS Jy ke TET S55) ee LO) 
JE OE ACRE BN alee 58 Uae He tN 


would be independent of each other. 

This system contains 3s — 8 invariants. ‘The groups 
potie -Smetiberedys POInts aa. We eyo 1a, oh a Pe 
would have just 3s — 8 independent invariants, for which 
we can consequently take those of the system (9), provided 
the determinant J is not equal too. A sufficient condition 


for any one of the groups that it may possess the second 
property is thus that the determinant must not vanish. 
The invariants /(1,2,3) for the different groups are as 
follows: — 
Group A. 
TCA YP 
{ Ly, ees the a AGS ie 454 . Ly, ueAy or 2,(x, a x5) | . 
L”, Mae dais 2,(4, at x;,)] Lys nite Bikes 2, (x, «ie x;) | 


[4 aot ae 2,(4, nec A))| \ 


( 


[yn TE Deere, Z(4) he x;)| 


Group B. 


7(T,253) <i } (11 watet (41 — X;) ea Gr — Jy) (tae x) f. 
eo ts (2 et %) 


Group I’. 
_ fof — ado + f(B — 2a)7 + a’ fio 
IEG OVID Foe {ee \. 
where a= 4%, — +, BP =%m%—2x,; p=—=F &% —4,)—a@ 
(4 peat 25), O14 (2 (ni a) ere B (2. -+ 25), Meraii4 On a ase 
a (4 + 2,). | 


Group A, 
1(1,2,3) = 
| (30°o — 3f'r + afp)(a—f) \ 
a’ (38 —a)o-— f (3a— f) t+ af (a+ B) py” 


with the same meaning for a, (, etc., as in the case of 
Group [., 


Group E. 


b) 


el as ee a (1 — 2) @ — a) (2 — 4) + 9) 


Xy— Xe cA a X3) (x, ra: X3) (2 ae 22) —- p\ 


where g = ("1 — y;) (x; — He Si A stay (Aiea 


Group Z. 
When 6 ts equal to 7, 


at! 


£8 : 
ee?) (n= a) a aye : 


When 6 is not equal to 1, 


7(1,2,3) ie Pia Soles Xo) (ry — 23) (4; Tash win ‘ ii f 


26 
a 


where a = (y, — Jo) (41 — %3) — (11 — Is) (41 — Xp), 
Ca — ax) (41 -— ¥8) — (2r — 25) 1 a). 


We find that D vanishes for the groups 4 and J’ and 
for the group Z when 6 = —1. Upon examination of the 
invariants of these groups we easily find that 


DEL 2A CAOIA Leal Adee a Lyees. 


in the case of group 4; 


bt 214) => 1 (253 4) Cay 0,4) =e 013 2,3) 


in the case of group 1’; 


and 


V G20} + V e308} + Vara) 
= Niet 


in the case of group Z when 6 = —1. 

As four points must have four independent three-point 
invariants'for the groups required, it is plain that the groups 
above must be omitted. Thus only the following satisfy 
the requirements: 


P,Q *9 XP — IG, VP, APT IGT, XP + IY 
TAT, IP IG /P 


BQ *g+7, 2¢o + axr, x9 + 32°77, vq + 27, 
xp + Ig, UP + 32y9 + (ee + 3y)r. 


PV UF Ty XQ, BR QAGAT, I eee ae 
x*h +- 2xXxyg ee 


P: QQ, 7, XG; AT, 2xp og -}- art ap — XV + Ete 
: ya + (ay + bz)r; a (6 —1) = 0, 
bis not equal to 0, ov —1. 


All the groups 1-4 are therefore such that three points 
have one invariant ‘within each of the groups, while s points, 
s>3, have no invariants independent of such three-point 
invariants. 

Moreover, any group in space of three dimensions that 
possesses these properties must be similar to one or other of 


ais Wace 


the groups 1-4. We must, however, not forget that we 
have hitherto regarded x, y, 2, as complex as well as real 
variables; if we restrict x, y, z, to be real variables, the 
group-types 1-4 are not sufficient. 

We have thus yet to determine all real groups similar to 
the groups I 
of the variables. 


4, by means of imaginary transformations 


GC THE REAL AZGROU PS “SIMILAR TO.VEEE 
HIGHT-MEMBERED GROUPS. 


The principles involved in the solution of this problem 
are clearly exhibited in the memorable article by Professor 
Lie, ‘‘Ueber die Grundlagen der Geometrie,” pp. 355-418 
of ‘‘ Leipziger Berichte’’ for 1890, before mentioned.” 

It is of importance for the following to note the systems 
of surfaces or curves that are invariant within the groups 
1-4. We verify without difficulty that the groups 1-3 
have one invariant system of curves, viz.: 


pI 


“== constant, ? 
y —— «¢ 
and that the group 4 has the two systems: 


° ‘6 


x == constant, } x == constant, | 
y eee ce \ ) oy 


when a = 0; if ais not equal to v0, then it has but the one 
system: + = constant, y — constant. 
In addition, the groups 2-4 have one invariant system of 


surfaces, namely: 
“ ==.constant. 


1) See in particular pp. 404-416. 


Before we proceed to determine all real groups similar to 
the groups 1-4 by means of imaginary changes of the 
variables, we shall determine the real groups similar to the 
group 4, as the result will be used later in this investigation. 

Let the variables of such a group be denoted by 4, 14, 2%. 
By means of substitutions of the form 


is GF, Cag odie 21) sts (ay, Va Bid Vee 2ECe, 
2 == ete sae 


— Pp, Py, etc., being vea/ functions of x,, 1, 2, and z denot- 
ing as usual |“-1, = the group 4 


Pg x9 + 7, tp — V9 — 227, Wp — 27, XP + VG, 
xp + wyg-+ (y— xz)r, ayp + y'¢ + Ay — x2)r, 


must be transformed into a real group. 

Let us fix a real; arbitrary point 2; —="a, 9) ==, eee 
within the sought group. The number of its independent, 
infinitesimal transformations will evidently be reduced to 
five, which will all be real, forming a five-membered group. 
To this group will correspond the subgroup of the group 4 
obtained by fixing the point corresponding to +7, = a, 
yi = 6,, 2, = ¢, and.determined by the equations 


a D(a, b,, C1) = ZP,(Q,, On oy ee a, V 7 etc. = b, 
Fagen el eae 
a, b, c being, in general, complex numbers. 
Now, substitutions in the group 4 ot the form 
we a = a Oe ae, 


leave that group unaltered, after we drop the primes. For 
the -poi1t corresponding to7 4, = a), 9. == Bj 12, == 
may therefore take the following 


DOH) 0 ius Gh a Ce ees 


cans Ss AY acces 


and by fixing this point within the group 4 we obtain the 
subgroup: 


De a Ore ee Ti eee 
Ep apg + (y— 22)r, yp ty gel — x2). 


This group is thus, by means of the substitutions (10), 
transformed into a read, five-membered subgroup of the 
sought group, which subgroup we shall call G’. 

It is easily verified that the group (11) has one and only 
one invariant system of surfaces, 

oe constant. 
1 

To this system must therefore correspond a real system 
belonging to G’ (an imaginary system could not belong to a 
real group unless the conjugate-imaginary system also 
belonged to that group). Let this system be given by 

Ji 


—— = constant, 
1 


so that we have a relation 


a ee 
x raves \Geae 


jf; being a real or a complete function. 


Having established this relation, we go back to consider 
the group 4. It has two invariant systems of curves, 


pee constant, l 2 == constant, \ 
Sa ; 


’ ph 66 
2 66 \ ae re Ane aes 
iy, Aise= 


The required group has therefore also two such systems, 
one of them being given by 


—— 40— 


J F A 
a = constant, 2(%1,.91,21) + 7p. In2) 
if 
= constant 25% ares 


If this system is not a veal system, 7. ¢., if @, is not 
identical with #(@,), the conjugate-imaginary system 


) 
Albee constant, Y, — z7p,=—= constant, 


“2 | 
A 


would also be invariant, and should therefere correspond to 
the system 


2 == constant, y — x2 = constant. 


That is, we should necessarily have 
ah = Ale, y — x2). 
But this relation, in connection with 
a Rp gee 
Bh 
would lead to the absurdity 


a = f(2, 7 na XZ), 


co 
Consequently, the system (12) can not be an imaginary 
system, and so we may designate it by © 


Os 


== constant, 2+, — constant. 
; 


Hence we infer that the variables x, y are changed into 
4, y,. Accordingly, the shortened group 


f= Ese DID AOA oye Eh 2 ee 


of the group 4 is changed into a shortened group 


Tye Gi (41, WP <a A (Hy Vid see Loree ie See 
of the required group. 


Professor Lie has considered all veal groups in two 
variables. By referring to his researches (see in particular 
chapter 19 of vol. III of his work ‘‘ Theorie der Transfor- 
mationsgruppen’’), we find that all real groups 


Vf = Ela A + nan yon? = Ds ie Co tas 


that are similar to the shortened group 


Siti Ce VPS Hil Be WG eee ey tak ay B 
of group 4 are represented by the same type. 


Using this general. type we have thus still to determine 
a, P, v, ete., as functions of 1%, ¥,, 2, so as to find all real 
groups of the form 


Oy ary Geer Br, AiG a Vai, Pi ha te O71, 
MNp~r t+ My, MiP; + 4 rime xy py - KiNG: + UN, 
MNP + WN =e AF, 

similar to group 4. 

Now, suppose that a, (, vy, etc., are so determined. We 
can evidently, by a process similar to that employed in 
simplifying the groups /’— Z, so determine a substitution 

fe SSN ee eee aa 
J(%, 7; 2) being a real or a complex function, that we get 
either one or other of the groups 4 or B. 


Since the required group or groups are similar to group 
A, we could get only this group in that way. It is then 


clear that the required group or groups are obtainable from 
group A by means of the inverse substitutions 


Bk, WSN 2S Dae oy) = ee 
The invariant system of curves 
2 = constant, vy — xz = constant, 
is thereby changed into 
a + 7f = constant, y; — ma —ix,f = constant 
The conj ene ncet eee system, 
a —7f = constant, y, — *1,@-+ 7x,6 = constant, 


must plainly be equivalent to this system; z. @., we must 
have the relations: 


eet ooo thy ety med Te Ci aL Ry 10,35 
Wy — Ha+ in p= yp jat iB, 1 — xa —ix,f}. 


Now, if a and f are independent of each other, so that we 
have no relation 


ACF 9) eo 
then we should have 


Xy oy PD, {ay Br, 4 Gare p ‘a, Bi, 
Or 


at BS TR) es 
which is impossible. Hence f(a, f)= 0, and we may put 


2 soley Jf B1)s SOL Se Seay, eee (13) 


The infinitesimal transformation xg + 7 of group A now 
becomes 


ug + ae + iy at ea” or HQ, fin. 


But then we get the superfluous infinitesimal transformation 


(Ay 
ig Nae Reel @) ees ater 


ae 


since the groups we are seeking must be real. 
The transformation (13) must therefore be of the form 


VMs ek lw 


Substituting throughout we finally find that £ = o, and the 
group 4 is left as it was in form, and written in the vari- 
BDIESE Ts) Vj, 26 

Accordingly, group A is a type of all real groups similar 
£0° 2h. 

‘Having considered this group at length we pass more 
rapidly over the groups 1—4, noting that we shall use much 
the same method of treatment. 


The groups 1, 2, and 3 have one invariant system of 


curves: x — constant, y= constant. 

All real groups similar to these must therefore also have 
one such system, which must be a veal system. For, were 
it imaginary, say 


a+ 7 = constant, vy -+ 76 = constant, 
then the conjugate-imaginary system, 


a — i — constant, vy — 70 = constant, 


eye aw mt 
Pee ‘ 


jo ae 


ses ig 
would also be invariant. But these two systeris should 
plainly be one and the same, there being only one belonging 
to the groups 1-3. That is, we should necessarily have the 
telations a — 78 = f(a+ 78, y+76); y —16 = f(a-+ ap, 
y+ 76); a, &, vy, 6 being functions of the independent 
variables *), v;, 2;. Now, these relations lead to absurdities, 
as is easily seen; unless the two systems are represented by 
the same real form 


g; — constant, ~, = constant, 
or, as we shall denote it, A 
x, == constant, vy, — constant. 


But then the variables x, y are transformed into 1, 1, 
so that the shortened groups X, f= &,(4, yp + 1, vq, 


i == 1, 2,....., 7, Of the groups 1-2 are transtonmecuenas 
shortened groups 
My Ei (X, W)~i ae Ni (Ks VN, = I, 2, ee | 7) 


of the required groups. 

By referring to Professor Lie’s researches on the real 
groups in two variables, above. mentioned, we find that all 
the real groups similar to the groups 


Af Ee gh Eas), © = 1p ey 


are represented by the same types. 

Proceeding now in the same way as we did in the case of 
the group last considered in determining the remaining 
terms ine i, 272 a ee , 8, of the infinitesimal 
transformations, we find that these terms are of the same 
form as the corresponding terms of the groups 1-3. 


The groups 1-3 stand therefore as types of all real groups 
similar to them. 


We come now to group 4: 


r, Xr, p,q XG, 2xP Ty + 27, 9 + (ay + ba)r, 
xp + xyqg + xer; (6 — 1)a = 0, and 2 is not equal to o, 
or — I. 


This group has one invariant system of surfaces, + — con- 
stant. The required real group or groups have therefore 
also one such system, say +, — constant, which is real. 
Hence we infer that a relation, real or complex, 


x = P(x), ig a 
exists, and that the shortened group x Viele) Xf ae 
2a J = x'p of the group 4 is changed into shortened groups: 


¥f= Liar Wee T5273) 
of the groups sought. 


Now, Professor Lie has shown that these groups Y,f 
=<; (%4)f1; 2 == 1, 2,3, can: always be.represented, by a 
group having the infinitesimal transformations 


Pis *if~i, 4p. 

The general transformation of the group produced from 
the infinitesimal transformations /, xp, x°p, being 

ax + 6 

cx + a’ 
should then, by means of the change of variable + = P(x,), 
be equivalent to the general transformation produced from 
the infinitesimal transformations ~,, +1f,, 2%), , which is 


ax, + 
Gtr Cy 


ap > _ am + bh 
cptd ~ax,+d,’ 


That is, 


2-469 


or y 
oF Gat, be 
atiSniee S82 oe. 


Now, a substitution of the form 


ig Maa Os eye Bs a 
Teed os dyn en eee bgt ge 


does not alter the form of the group 4, after we drop the 
primes. 
Hence it is obvious that we can put 


, fees ape ips ME 
Group 4 has also at least one invariant system of curves, 
a = constant, y = constant. 
If this is transformed into a real system, say 
+, = constant, v, = constant, 
the shortened group X,f—= &(x)p+ nlx,y)¢, 7 = 1, 2, 
ae , 6, of group 4 is transformed into a shortened group 


es Ei (X1)p~r + 1 OX, Vi) 915 (Bree Viggo GO : 6. 


of the required group. As in the case of the groups 
previously considered, the group 4 is then a type of all real 
groups similar to it. 

On the other hand, the system + = constant, y = con- 
stant may be transformed into an imaginary system, say 


a4 == constant,.y, 22) == constant: 
The system 
x, == constant, 4; =-"72;"== constant 


would then also be invariant, and would accordingly 
correspond to the system : 


«x = constant, 2 ==constant, 


ne <j 


which is invariant only when a = 0 (see page 37). 
The variables 7, y, z are therefore changed into 1, 1, 2, 
according to the relations 


A 4 We ae fo we tZ15 Q(X, 2) =f, — 04, | 


qT I 
ST oA ee eG ar a a ee) 


The group 4 contains the infinitesimal transformation g. 


. This is changed into 


I 5 
AC; ati 2), 


so that the required group contains the infinitesimal trans- 
formations 


O15 Vy. 


Now, if we transform g, back into the group 4 by means of 
the relations (14), we get 
I 
Y aa 
Oz 
This must be expressible as a sum of the infinitesimal 
transformations of group 4 with constant coefficients. 

Hence, 

I 
— =a + ax, 
Op 

Se 
so that 


a g SE eae 
p= 7 gli 2 aes + p(x). 
Again, +g becomes %(%x,g, — 7%\7,), giving the infinitesimal 


transformations 
: MiQiy AM 


Transforming 2,7, back into group 4 as we did g, above, 


we get . 
xg —-+- Xai, == Ast )r: 
We therefore conclude that a, = 0. 


The infinitesimal transformation / is changed into 


Bae ee Teg, + 5 eer, 


Let (x) = a(v;) + 78(%), and this last infinitesimal 
transformation breaks up into the two 


I 0@ I of 
Pe Gy Mh» FQ, ah ar, 


gO, 2 0x, 


Now, it is evident that the three infinitesimal transforma- 
tions: 


2ap + 99 + 27, yg + ber, xp + xygq + x2r 


would give rise to at least three such belonging to the 
sought group, independent of each other and of those 
already determined, 


I 0@ 1 0f 


GWi5 M13 MMs Hy A 5 dxf! 20x)? Bq, + or. 


But that would give us nine infinitesimal transformations 
in all, unless those last written are not all independent. It 
thus appears that we should have the relation 


BQ > ar, = Ca Bx,)qy + C6 ae Dx)n, 
A, B, etc., being constants. 


This gives 
(x) = A’+ B's, 


and the relations (14) are therefore the following: 


FS Ay. Yrs yi aay), wit a teen ia; 


I 2 ; , 
or Cain Nea 8 a Rane seet cit 2); 


I ys, f ; 


4, => 
2 


The remaining infinitesimal transformations of the group 
4 are now without difficulty changed into the following: 


201 Pr + NG + AN, XP Ig: Ae, 
{ie bho te, OF 
{ y,2( —i+6)+ 2401+ b) hin, 
when we take account of those already determined: 9,, 7, 


A915 AY, Pi- 
If now 6 = m + 7m, the infinitesimal transformation 


fant + 4) + ta(r — 4)} a + 
{yd (— 1+) +a +d}n 


breaks up into the two 


(i + m) (ng ani) + 2 in — nd, 2Cng + 217) 
Bie 20) eds Vi) 3 


which must not be independent, in order that we should 
not have more than eight infinitesimal transformations. 
Accordingly, 

Tis 9707 ab i9e" 
and, since 4 is not equal to 0, or — I, we may not have 
// ee Ee Camels 


The only real group that we have thus obtained other 
than the groups 1-4, page 36, is the following (omitting the 


subscripts, and putting c = 


ay. 
1+ m’** 


| VS Sh Ds Gy KG, 2k PA VG, a erg 
yO + ar + clzg— yr), xp + xyQ -- xer. 


The real, eight-membered groups are now completely 
determined, and we can therefore give the following result: 

All the real, eight-membered groups satisfying the require- 
ments given on page 5, are similar to the groups 1-5 by 
means of a veal transformation of the variables. The 
two groups 4 and 5 are similar through an imaginary 
transformation. 


D. SOME GEOMETRICAL PROPERTIES OF THE 
EIGHT-MEMBERED GROUPS. 


Since two points have no invariants, the motion of a 
point in general position would not in any way be restricted 
if we fix any given point 7,, 7,, 2,. All the groups being 
imprimitive, we should, however, expect that certain sys- 
tems of points or some isolated points would be restrained 
to less than three degrees of freedom, when a given point is 
fixed... Thus the points on the line <== 35), 3 =e 
fixed point being x,, y, 2)), have only one degree of freedom, 
as they must remain on that line. ‘This follows at once from 
the fact that all the groups 1-5 have the invariant system of 
curves (lines) a = constant, > 7 ==: constant; 7, 2it ese 


curves are interchanged by the transformations of the 
2ToUups. 

Since three points are connected by one invariant rela- 
tion, a point in general position is free to move only on 
some one surface, when two points, also in general position, 
are fixed. This surface we shall with Professor Lie call a 
Pseudosphere, and the two fixed points we shall call its focz. 

To every pair of foci corresponds a one-fold infinity of 
Pseudospheres, given by the equation 


1A Mi it Vey 235 XV, &) == constant, 


I(%, try 213 ete.) being the invariant in the three points 
Bi ie Ae ey, ye. +( We stppose the two fixed 
POLBES EO De) °9/5° S15 25,4; So) 

As the invariant / for the different groups are all given 
(pp. 34, 35) except in the case of group 5, we can thus with- 
out difficulty write down the equations of the Pseudospheres. 


2g 
3 3a 
Let us now consider group 1. If we use e >” fora new a 
we change 7 into %<27, and the invariant 7 becomes 


| Deed a Hs) ON I) Se | 
| Eee 
The Pseudosphere 


Se ie ea 
Oa Veet Ths 


(68 


is accordingly a plane. When the foci are any two points 
lying in the plane 
ax + by —1= a, 


the Pseudosphere becomes 


ax + by — 1 = cz, 


Thus it appears that all Pseudospheres with the same 
pair of foci form an axial pencil of planes, and any given 
Pseudosphere has for foci any point-pair in a given plane. 

The last remark is also evident from the fact that the 
group is ‘‘systatic,’’” 7. ¢.,-when any point x,, 7), 2) 15 
fixed, then is every other point on the line 


Pe RG AMET TY 
also fixed, which is seen directly from the infinitesimal 
transformations of the five-membered group obtained by fix- 
ing the point 41, "1, 21: 
(w— 41)9, (4 — 1) — (9¥— Ni), (9 — IP, 
(4 21)"P + (4 — ma) (9 — NG +(e er, 
(7 — 11) (¥ — Wn) PY = Id TAY See 
Therefore, by fixing the two points 11, 1, 21; 4), Vo; 2o5 
all the points on both lines 2 =, 9 = 3152 =e 
y= jy, Yemain invariant; and accordingly also every 
straight line joining these two lines, since the group is 
projectivé, as is easily seen. © 
It is therefore clear that every point in the plane of the 
EWO? LIES (A ey AY ig : Y= yl, must be fixer 
when the. Wo, Poms B15. Vax 2 tga ay ae eee eye 


The Pseudosphere of either of the groups 2 or 3 is a 
ruled surface of degree 3. “The generators are parallel to 
the plane + = 0. ‘The foci lie on the Pseudosphere. 


In the case of group 4 the Pseudosphere is transcendental 
or of degree 24. We shall accordingly consider the group 
only for special values of @ and 8. 

Let 6— 1, anda—o. ‘Then the invariant 7 becomes 

C21 2) Gi = 4%) — (i — 4%) a = %) 


Oe 25 2) ar, On EI (xy ny X,) wer On Sg) Gy 7 x,) 


1) ‘‘Systatisch.’? See p. 386 of ‘‘ Leipziger Berichte’’ for 1890. 


Seay eee 

The Pseudosphere is a plane passing through the two 
fixed points. Thus it appears that for any given Pseudo- 
sphere we may take any pair of points on a given line as 
foci. We should therefore expect that when two points are 
fixed, any point on the straight line joining them would be 
restricted to move on that line. ‘This is also evident from 
the fact that the group is projective. 


E. THE SEVEN-MEMBERED GROUPS. 


The groups of the class we shall now consider have two 
invariants for three points. These groups we can determine 
in a manner similar to that which we employed in the case 
of the eight-membered groups. We find, however, a great 
maty more groups, principally owing to the fact that if a 
group here has an invariant system of surfaces + = con- 
stant, so that it is of the form: 


ee ey, 2g GAY e)Y, 2 SRT, Sy oy Psy 
the shortened group, 
et = e(0)p, 


need not be three-membered, as in the former class of 
groups. It must, however, be at least two-membered, in 
order that the two independent three-point invariants may 
not be functions of x,, x,, x, only. For, if that were the 
case, the invariants of s points, being, according to defini- 
tion, made up of the three-point invariants, would all be 
rreeqiront the VATA Dlesi vip, Vs y06 i c/n 5, Vek Sry Sano Maw es Sen LEMS 
would lead to a contradiction, as we saw on page 18, the 
groups being finite. 


Se a 
We would find that every group of this class satisfying 
the definition given on page 5 would be similar to one or 
other of the types given below. ‘The classification is, as in 
the case of the eight-membered groups, based upon the 
geometrical figure left invariant by the transformations of 
the respective groups. 


A. Primitive Groups. 


Here we have the previously mentioned Group of Euclid- 
ean Motions and Similar Transformations: 


Ll. 1B, 9, % 2Q— It, 2D: — KV, VP — 29, 2. - Vg ee 


B. ILmprimitive Groups. 


a. ' One invariant system of curves: Y 


Bit IFAT, XO + UXT, tf — IG; IP + VI, 
xp + vq + 22r. 


Bx Dy Gy Vy XQ) 4D 99, YP Bi ee 


Ps XO +7, HQ 2x7, WG 34°, 2xP + 3y9 7 BP, 
xp 4- 3ayg + (ve + 3y)r. 


Bo, xg tr, eq + 2xr, yg + 27, xp — 27, 
xp + 2ayg + 2970. 


EO; 


p iS 


ze 


13, 


14. 


-, gq, VY, XQ, 9 a AY, xp “Fig, 
xp 2xyg + (ax + y)r. 


Pe Ty AE LIT, IG 227) Bap Vg, 
xp + «yg + Yey'r. 


Dy Gielen LO OL BY te VI, 
ap + kyg + [2(k — 1)2 + ax’*]r, &— 2) a=—o0. 


LE GI LG AT NT NSE ay)r, xp byr. 


BW 1% HY X’7, xr* a’g + (%2x*° + ay)r, 
xp + 399 + 42r. 


Ps Uy %, x9 + yr, xr, yg + 227, xp — ar. 


Ty) 27, Pg XT, XY xT, HG (hat + ay), 
xp + avg + 62r. 


Two invariant systems of curves: 


BG, 9+ 1, e+ 227, x°¢ + 32°7, x'¢ + 42°F, 
ap + kyg + [(# — 1)2 + alr; (k— 1)a = 0, 


DG xg +7, ag + 2xr, 9 + 34°, 2g + a'r, 
xp + (sy + )9 + Ge + 5r'yr. 


oa d ! git Ye tatoo oy ee 
7 | , fe eS es 
se Pp; q; ag+r, xg + 2x7, Ja A ee 
‘i 7 Ad ie ae xp — er. 
a | PD *9 X97, XG + 347, VOT 27, Hp ae 
PO 29,4 7a 7; xg + 347, xg + 6x°r, a 
7 xp a (57 -+ x°q) + (32 at 10x*)r. ogy 
18 Bs 9 49, PG +7, 74307, 09 + 6er 
. xp + keyg + [hk — 2)2 + alr ere 
Pi 5% xq, xq + ar, Og +327, ae 
19. | ae 
4 xp + kyg + (A — 1)2 + ax]r; (k — 2)a = 0 
DP q; 1%, XQ, xg + i 6 x9 oe By, 
20. | ren 
tp Gy + 49 4 eee 
21. DU 1 49, ¥"¢ + xr, yg + er, xp — are : : 
22 Dp; g YT, XQ, AY, X " + aZr, xp a Sy oT bs 
a is not equal to o. hk 
Re x4, ar, yg azr, xp ed ber; ‘oe 
23. 
a a is not cue) to o. 
ta ' a 2 | * wh ¥ rope 
i oe, WO ee eer ie 
% * y 4 ; . 4 . ae & 
4 ee aad jas : i 4 Me s3 
hy ; 


24. | P: J, 7, XY, AT, VO ae Aa, Xp + (22 ne ai lee | 


c. A o' invariant systems of curves: 


Oe 2, g, %, xg, xr, 2xp + yg + 27, b+ xyq + x2r. 


d. ‘Two invariant systems of surfaces: 


20; Pip, py ae PA ad 5 VT, VE I. | 
27. | VY, XT, Pp xp, 7, V9, vg TIT | 
28. | Ket VF 4 Bi xp aE azZr, J, 2V oe ar, vq + V2r. | 


We shall not investigate these groups in detail nor shall 
we attempt to determine the canonical forms of all the real 
groups similar to them. 

It might be of interest, however, to find whether there 
would be any veal primitive groups besides group I (7. é., 
groups that do not possess any real invariant systems of 
curves or surfaces), belonging to this class. This problem 
we shall proceed to solve—in fact, we shall consider the more 
general problem ‘‘to find all the real-primitive groups in 
three variables that are not similar to the primitive groups.’’ 


F, THE REAL-PRIMITIVE. GROUPS IN THREE 
VARIABLES. 


A real-primitive group in three variables might leave an 
imaginary system of surfaces invariant. But it is evident 
that it must at the same time leave the conjugate-imaginary 
system invariant. Let then the two systems be represented 
by 

x1 + ty, = constant; x, — zy, = constant. 
Therefore the system of curves 
Ky + 2y, == 1eonstant, 41)=— 277) == constant; 
is invariant, which system, being equivalent to 


x; = constant, 7, = constant, 


is a real system. ‘The group could therefore not be real- 


primitive, and we are thus led to the conclusion that a veal-_ 


primitive group in three variables can have no invariant 
system of surfaces. 


The next question to serene is whether such a group 
can have an invariant system of imaginary curves, say 
a-—-7z@ = constant, y -+ 26 = constant, 
so that it would also have the system 
a-— if = constant, vy — 70 = constant. 


These two systems could not be identical unless they were 
real (cf. page 44). 

Hence, a real-primitive group in three variables must have 
at least two invariant systems of curves, if any. 


i ae 


All imprimitive groups in three variables having no 
invariant system of surfaces have been determined by 
Professor Lie.” By examining these we find that one 


group only has two invariant systems of curves, namely 
the group 


P; q) a ay r; Xp fey Gi ees Boe 2"7, Xp + Id; 
xp + xyg + (y — x2)r, xp + 99 + 2(y — x2)r. 


We have already examined this group (pp. 38-43) and. 
found no real-primitive group similar to it. Accordingly: 

There are no real-primitive groups in space of three 
dimensions except those that are fully primitive, 7. e€:, have no 
invariant systems of curves or surfaces, real or complex. 


We now go back to the seven-membered groups. Ac- 
cording to the preceding analysis, there are no _ real- 
primitive groups belonging to this class except group 1 
(page 54) and the groups similar to this group. | 

Group 1: 


+S GY AOR Ai De 


zp, ip — x9, xp +99 + er 
leaves invariant the differential equation of the first order 


ax’ + dy’ +- dz’ = o, 
and this one only. In other words, there is a cone of line- 
elements of the second degree connected with every point 
x, ¥, 2; which cone in this case is imaginary. But it may 
be transformed into a real cone and can as such be repre- 


sented by the equation 


ax* + dy’ — dz’ =o. 


1) ‘‘ Theorie der Transformationsgruppen,’’ Vol. III, chap. 8. 


The corresponding group: 


4 


Ba ror +20, 27+ 2p, 29 — Id, 


Re, xp +99 er 


¢ 


neleneiae to this ee 


io G. THE SIX- MEMBERED GROUPS. 


The groups belonging to this Glass are very numerous, 
being all the groups in three variables for which two points i 
have no invariants, and for which Ea points: have thre« 
(cf. page 11). RS ea 

- There are no primitive groups contained i in ‘this class, and 
therefore no real-primitive (cf. page 59) 


Nd 


6 1 —— 


Wi Yeu 


I, the undersigned, was born at Illerup, Denmark, Janu- 
ary oth, 1873. I attended the public and private schools of 
Copenhagen until 1888, in the spring of which year I 
passed the ‘‘ Almindelige Forberedelses Examen” at the 
University there. In the fall of the same year I went to the 
United States of America, where I have resided since. I 
became a citizen of that country by virtue of my father 
becoming a citizen while I was yet a minor. 

After having followed the profession of civil engineering 
for some time, I entered the Leland Stanford Junior Univer- 
sity, California, in the fall of 1894. There I devoted 
myself principally to the study of Mathematics under the 
guidance of Professors Allardice, Little and Green. The 
degree of Bachelor of Arts was conferred upon me in the 
spring of 1896, and that of Master of Arts in the spring of 
1897. In the fall of 1897 I entered the. University of 
Leipzig. Here I have remained since, and have attended 
lectures given by Professors Lie, Mayer, and Drude; semi- 
nars by Professors Lie, Mayer, and Engel, and laboratory 
work in Physics under Professor Wiedemann. 

ne ba DICHEELDT: 


